Let $R$ be a left Artinian ring. Dlab and Ringel have shown that $R$ is hereditary if and only if every chain of idempotent ideals can be refined to a heredity chain . In particular, if $R$ is a basic hereditary ring, then every primitive ideal is a heredity ideal. The converse to this is clearly false. (See Example 1). We will introduce a class of rings that includes serial rings and monomial algebras, for which the converse does hold.
"Primitive Heredity Ideals." Missouri J. Math. Sci. 13 (1) 36 - 42, Winter 2001. https://doi.org/10.35834/2001/1301036